Vol. 4, No. 6, 2010

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Integral trace forms associated to cubic extensions

Guillermo Mantilla-Soler

Vol. 4 (2010), No. 6, 681–699
Abstract

Given a nonzero integer $d$, we know by Hermite’s Theorem that there exist only finitely many cubic number fields of discriminant $d$. However, it can happen that two nonisomorphic cubic fields have the same discriminant. It is thus natural to ask whether there are natural refinements of the discriminant which completely determine the isomorphism class of the cubic field. Here we consider the trace form as such a refinement. For a cubic field of fundamental discriminant $d$ we show the existence of an element ${T}_{K}$ in Bhargava’s class group such that ${q}_{K}$ is completely determined by ${T}_{K}$. By using one of Bhargava’s composition laws, we show that ${q}_{K}$ is a complete invariant whenever $K$ is totally real and of fundamental discriminant.

Keywords
integral trace forms, cubic fields, Bhargava's class group, discriminants of number fields
Mathematical Subject Classification 2000
Primary: 11E12
Secondary: 11R29, 11R16, 11E76
Milestones
Received: 18 June 2009
Revised: 5 December 2009
Accepted: 15 May 2010
Published: 25 September 2010
Authors
 Guillermo Mantilla-Soler Department of Mathematics University of Wisconsin-Madison 480 Lincoln Drive Madison, WI 53705 United States